Startseite Mathematik General theory of Caputo-type fractional differential equations
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General theory of Caputo-type fractional differential equations

  • Kai Diethelm
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Volume 2 Fractional Differential Equations
Ein Kapitel aus dem Buch Volume 2 Fractional Differential Equations

Abstract

This article describes the fundamentals of the theory of ordinary fractional differential equations of Caputo’s type. Starting from the existence and uniqueness of solutions and the well-posedness in general, the flow of topics continues via a derivation of explicit solution formulas for certain important classes of problems and the discussion of their smoothness properties to the stability properties of these solutions. The main focus is on initial value problems, but terminal value problems are briefly considered as well. In addition to dealing with standard single-order problems, the presentation also contains a short discussion of multiterm equations and multiorder systems.

Abstract

This article describes the fundamentals of the theory of ordinary fractional differential equations of Caputo’s type. Starting from the existence and uniqueness of solutions and the well-posedness in general, the flow of topics continues via a derivation of explicit solution formulas for certain important classes of problems and the discussion of their smoothness properties to the stability properties of these solutions. The main focus is on initial value problems, but terminal value problems are briefly considered as well. In addition to dealing with standard single-order problems, the presentation also contains a short discussion of multiterm equations and multiorder systems.

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. General theory of Caputo-type fractional differential equations 1
  5. Problems of Sturm–Liouville type for differential equations with fractional derivatives 21
  6. Maps with power-law memory: direct introduction and Eulerian numbers, fractional maps, and fractional difference maps 47
  7. Symmetries and group invariant solutions of fractional ordinary differential equations 65
  8. Operational method for fractional ordinary differential equations 91
  9. Lyapunov-type inequalities for fractional boundary value problems 119
  10. Fractional-parabolic equations and systems. Cauchy problem 145
  11. Time fractional diffusion equations: solution concepts, regularity, and long-time behavior 159
  12. Layer potentials for the time-fractional diffusion equation 181
  13. Fractional-hyperbolic equations and systems. Cauchy problem 197
  14. Equations with general fractional time derivatives–Cauchy problem 223
  15. User’s guide to the fractional Laplacian and the method of semigroups 235
  16. Parametrix methods for equations with fractional Laplacians 267
  17. Maximum principle for the time-fractional PDEs 299
  18. Wave equation involving fractional derivatives of real and complex fractional order 327
  19. Symmetries, conservation laws and group invariant solutions of fractional PDEs 353
  20. Fractional Duhamel principle 383
  21. Inverse problems of determining sources of the fractional partial differential equations 411
  22. Inverse problems of determining parameters of the fractional partial differential equations 431
  23. Inverse problems of determining coefficients of the fractional partial differential equations 443
  24. Abstract linear fractional evolution equations 465
  25. Abstract nonlinear fractional evolution equations 499
  26. Index 515
https://doi.org/10.1515/9783110571660-001

Kapitel in diesem Buch

  1. Frontmatter I
  2. Preface V
  3. Contents VII
  4. General theory of Caputo-type fractional differential equations 1
  5. Problems of Sturm–Liouville type for differential equations with fractional derivatives 21
  6. Maps with power-law memory: direct introduction and Eulerian numbers, fractional maps, and fractional difference maps 47
  7. Symmetries and group invariant solutions of fractional ordinary differential equations 65
  8. Operational method for fractional ordinary differential equations 91
  9. Lyapunov-type inequalities for fractional boundary value problems 119
  10. Fractional-parabolic equations and systems. Cauchy problem 145
  11. Time fractional diffusion equations: solution concepts, regularity, and long-time behavior 159
  12. Layer potentials for the time-fractional diffusion equation 181
  13. Fractional-hyperbolic equations and systems. Cauchy problem 197
  14. Equations with general fractional time derivatives–Cauchy problem 223
  15. User’s guide to the fractional Laplacian and the method of semigroups 235
  16. Parametrix methods for equations with fractional Laplacians 267
  17. Maximum principle for the time-fractional PDEs 299
  18. Wave equation involving fractional derivatives of real and complex fractional order 327
  19. Symmetries, conservation laws and group invariant solutions of fractional PDEs 353
  20. Fractional Duhamel principle 383
  21. Inverse problems of determining sources of the fractional partial differential equations 411
  22. Inverse problems of determining parameters of the fractional partial differential equations 431
  23. Inverse problems of determining coefficients of the fractional partial differential equations 443
  24. Abstract linear fractional evolution equations 465
  25. Abstract nonlinear fractional evolution equations 499
  26. Index 515
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Heruntergeladen am 2.1.2026 von https://www.degruyterbrill.com/document/doi/10.1515/9783110571660-001/html
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